Issuing banks typically add an FX fee to the cardholder for cross border transactions. Not always, but many times they do.
This is misleading. Banks put a fee on foreign transactions, many do this even if the transaction is in the card's native currency.
The card companies always add a margin to FX conversions, usually in the 0.5% range. This is fairly benign since the market rate can move between the transaction and the settlement, so sometimes you save money.
Even if the FX is notionally "free" they pick the rates. So they can set those rates to generate exactly the same profit for them as with fees.
You will probably not see rates that are even competitive with a dodgy FX cash place at an airport, let alone with the numbers you've seen on financial networks for what FX transactions by banks cost, but you feel happy because there was "no fee".